There are three main theorems about roots of polynomial equations:
Rational Root Theorem
This theorem is a way to find a root of a polynomial equation. One way to find it is to guess and check and that is what the rational root theorem does while minimizing the number of guesses and possible roots.
Conjugate Root Theorem
The irrational numbers a + √b and a - √b are conjugates . If a complex number or an irrational number is a root of a polynomial equation with rational coefficients, so is its conjugate.
Descartes' Rule of Signs
The French mathematician recognized a connection between the roots of a polynomial equation and the + and - signs of the standard form.
Practice Problems
1. What are the rational roots of 15x3-32x2+3x+2=0 ? Solve using the rational root theorem.
2. A quartic polynomial function P(x) has rational coefficients. If √2 and 1 + i are roots of P(x) = 0, what are the other two roots?
3. What is a third-degree polynomial function y = P(x) with rational coefficients so that P(x)=0 has roots -4 and 2i?
4. What does Descartes' Rule of Signs tell you about the real roots of x3-x2+1=0?
